Dipartimento di Matematica

Asymptotic planar N-bubble

Abstract. We consider in the plane a fixed finite number N of "bubbles", that is
disjoint finite perimeter sets which possibly share portions of their
boundaries, and look for configurations that minimize, under a volume
constraint, the total weighted length of their boundaries: the interface
between each bubble and the exterior is given weight 1 while the interface
between any two bubbles is given weight $2-\varepsilon$. We are interested in
the case when $\epsilon$ converges to 0: we prove that minimizing
configurations approach in the limit a configuration of disjoint disks
which maximize the number of tangencies among them. Moreover we obtain
some information about the structure of minimizers for small $\varepsilon$.